Method and system for key exchange and method and apparatus for reducing parameter transmission bandwidth

ABSTRACT

The embodiments of the present disclosure disclose a method and apparatus for reducing the parameter transmission bandwidth. The parameter sender reduces the values of the parameters before sending the parameters to the parameter receiver. This scheme reduces the bandwidth consumed during parameter transmission, thus makes the transmission more efficient. The embodiment of the present disclosure also discloses a method for key exchange. This method reduces the values of the transmission parameters before sending the transmission parameters. This saves the bandwidth compared with the protocol in the prior art. Besides, the embodiment of the present disclosure discloses a system for key exchange. The parameter sender sends the transmission parameters to the bandwidth processing unit. The bandwidth processing unit performs a modulo operation on the received transmission parameters and then sends the processed transmission parameters to the parameter receiver, thus reducing the bandwidth consumed in the transmission of transmission parameters.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of International Application No. PCT/CN2007/070944, filed on Oct. 23, 2007, titled “Method and System for Key Exchange and Method and Apparatus for Reducing Parameter Transmission Bandwidth”, which claims the priority of Chinese application No. 200610167172.3, filed on Dec. 15, 2006, titled “Method and Apparatus for Reducing Parameter Transmission Bandwidth and Method for Key Exchange”. The contents of the above identified applications are incorporated herein by reference in their entirety.

FIELD OF THE INVENTION

The present disclosure relates to network security technologies, and in particular, to a method and apparatus for reducing the parameter transmission bandwidth and a method and system for key exchange.

BACKGROUND OF THE DISCLOSURE

Some security protocols in network security technologies specify that entities that establish communications need to transmit some critical parameters. For example, the password-based key exchange authentication protocol for improving Diffie-Hellman key exchange specifies that entities that establish communications need to transmit parameters.

The purpose of Diffie-Hellman key exchange is to allow two users to exchange a key securely for future message encryptions. The security of Diffie-Hellman key exchange depends on the difficulty of computing discrete logarithms on finite fields. This algorithm provides full forward confidentiality, but is vulnerable to man-in-the-middle attacks.

The password-based key exchange authentication protocol is designed to alleviate man-in-the-middle attacks for Diffie-Hellman key exchange. The password authentication protocol depends on a poor and hidden shared secret and provides a specific mechanism for mutual authentication. Therefore, this authentication protocol can prevent offline dictionary and man-in-the-middle attacks.

The password-based key exchange authentication protocol has two communication entities A and B. Entities A and B share a secret password PW and public Diffie-Hellman parameters p and g. These parameters meet the following conditions: p is a secure prime number, making it difficult to compute discrete logarithms; the integer g is a primitive root of p and its step in Z_(p)* is p−1, that is,

g ^(P−1)=1 mod p.

FIG. 1 shows the process of key exchange as specified by the password-based key exchange authentication protocol in the prior art. The process includes the following steps:

Step 101: Communication entity A selects a random number R_(A) to compute the transmission parameter by the formula X=H₁(A_(A)|B_(B)|PW)·(g^(R) ^(A) mod p), and then sends transmission parameter X to communication entity B.

A_(A) and B_(A) belong to communication entities A and B respectively and are parameters known to entities A and B, such as addresses of entities A and B.

| is a concatenation symbol. For example, a|b indicates the concatenation of bit strings a and b.

mod is the modulo operator. For example, a mod b indicates the smallest non-negative remainder after positive integer a is divided by positive integer b.

H_(i)(u) indicates the hash function. It is a function used to compress bit string u in any length into a message digest in a fixed length. i is equal to 1, 2, 3 or any other natural number. Different i subscripts indicate different random functions that are mutually independent.

Step 102: Upon receipt of transmission parameter X from communication entity A, communication entity B checks whether X is 0. If yes, communication entity B terminates the process; otherwise communication entity B executes step 103.

Step 103: Communication entity B computes

$\frac{X}{H_{1}\left( {A_{A}{B_{B}}{PW}} \right)}$

to obtain g^(R) ^(A) mod p, and selects a random number R_(B) to compute transmission parameter Y and identity authentication parameter S₁. Finally communication entity B sends Y and S₁ to communication entity A. The formulas below are used to compute transmission parameter Y and identity authentication parameter S₁:

Y = H₂(A_(A)B_(B)PW) ⋅ (g ^(R_(B))mod  p) $S_{1} = {{H_{3}\begin{pmatrix} \left. {A_{A}{B_{B}}{PW}{\frac{X}{H_{1}\left( {A_{A}{B_{B}}{PW}} \right)}}g{\,^{R_{B}}{mod}}\mspace{14mu} p} \right| \\ \left\{ {\left\lbrack \frac{X}{H_{1}\left( {A_{A}{B_{B}}{PW}} \right)} \right\rbrack^{R_{B}}{mod}\mspace{14mu} p} \right\} \end{pmatrix}}\mspace{31mu} = {H_{3}\left( {A_{A}{B_{B}}{PW}{{g{\,^{R_{A}}{mod}}\mspace{14mu} p}}g{\,^{R_{B}}{mod}}\mspace{14mu} p} \middle| \left\{ {\left\lbrack g^{R_{A}} \right\rbrack^{R_{B}}{mod}\mspace{14mu} p} \right\} \right)}}$

Step 104: Upon receipt of transmission parameter Y from communication entity B, communication entity A checks whether Y is 0. If yes, communication entity A terminates the process; otherwise communication entity A executes step 105.

Step 105: Communication entity A checks whether identity authentication parameter S₁ sent from communication entity B is valid. If yes, communication entity A executes step 106; otherwise communication entity A terminates the process.

Communication entity A checks identity authentication parameter S₁ by using the following method: compute

$\frac{Y}{H_{2}\left( {A_{A}{B_{B}}{PW}} \right)}$

to obtain g^(R) ^(B) mod p, and then check whether S₁ is equal to the computation result of formula 1. If yes, S₁ is valid; otherwise S₁ is invalid. The following shows formula 1:

$\begin{matrix} {{H_{3}\left( {{A_{A}{B_{B}}{PW}{{g^{R_{A}}{mod}\mspace{14mu} p}} \frac{Y}{H_{2}\left( {A_{A}{B_{B}}{PW}} \right)}}\left\{ {\left\lbrack \frac{Y}{H_{2}\left( {A_{A}{B_{B}}{PW}} \right)} \right\rbrack^{R_{A}} {mod}\mspace{20mu} p} \right\}} \right)} = {H_{3}\left( {A_{A}{B_{B}}{PW}{{g^{R_{A}}{mod}\mspace{14mu} p}}g^{R_{B}}{mod}\mspace{14mu} p\left. \left\{ {\left\lbrack g^{R_{B}} \right\rbrack^{R_{A}}{mod}\mspace{14mu} p} \right\} \right)} \right.}} & (1) \end{matrix}$

Step 106. Communication entity A computes shared key K and identity authentication parameter S₂ and sends S₂ to communication entity B. The formulas below are used to compute shared key K and identity authentication parameter S₂:

$\begin{matrix} \begin{matrix} {K = {H_{5}\begin{pmatrix} {{A_{A}{B_{B}}{PW}{{g^{R_{A}}{mod}\mspace{11mu} p}}\frac{Y}{H_{2}\left( {A_{A}{B_{B}}{PW}} \right)}}} \\ \left\{ {\left\lbrack \frac{Y}{H_{2}\left( {A_{A}{B_{B}}{PW}} \right)} \right\rbrack^{R_{A}}{mod}\mspace{14mu} p} \right\} \end{pmatrix}}} \\ {= {H_{5}\left( {{A_{A}{B_{B}}{PW}{{g^{R_{A}}{mod}\mspace{14mu} p}}g^{R_{B}}{mod}\mspace{14mu} p}\left\{ {\left\lbrack g^{R_{B}} \right\rbrack^{R_{A}}{mod}\mspace{14mu} p} \right\}} \right)}} \end{matrix} & \; \\ \begin{matrix} {S_{2} = {H_{4}\begin{pmatrix} {{A_{A}{B_{B}}{PW}{{g^{R_{A}}{mod}\mspace{14mu} p}}\frac{Y}{H_{2}\left( {A_{A}{B_{B}}{PW}} \right)}}} \\ \left\{ {\left\lbrack \frac{Y}{H_{2}\left( {A_{A}{B_{B}}{PW}} \right)} \right\rbrack^{R_{A}}{mod}\mspace{14mu} p} \right\} \end{pmatrix}}} \\ {= {H_{4}\left( {{A_{A}{B_{B}}{PW}{{g^{R_{A}}{mod}\mspace{14mu} p}}g^{R_{B}}{mod}\mspace{14mu} p}\left\{ {\left\lbrack g^{R_{B}} \right\rbrack^{R_{A}}{mod}\mspace{14mu} p} \right\}} \right)}} \end{matrix} & \; \end{matrix}$

Step 107: Communication entity B checks whether identity authentication parameter S₂ sent from communication entity A is valid. If yes, communication entity B executes step 108; otherwise communication entity B terminates the process.

Communication entity B checks identity authentication parameter S₂ by using the following method: obtain g^(R) ^(B) mod p according to step 103, and then check whether S₂ is equal to the computation result of formula (2). If yes, S₂ is valid; otherwise S₂ is invalid. The following shows formula (2):

$\begin{matrix} {{H_{4}\left( {{A_{A}{B_{B}}{PW} {\frac{X}{H_{1}\left( {A_{A}{B_{B}}{PW}} \right)}}g^{R_{B}}{mod}\mspace{14mu} p}\left\{ {\left\lbrack \frac{X}{H_{1}\left( {A_{A}{B_{B}}{PW}} \right)} \right\rbrack^{R_{B}} {mod}\mspace{20mu} p} \right\}} \right)} = {H_{4}\begin{pmatrix} {A_{A}{B_{B}}{PW}{{g^{R_{A}}{mod}\mspace{14mu} p}}g^{R_{B}}{mod}\mspace{14mu} p} \\ \left\{ {\left\lbrack {g^{R_{A}}{mod}\mspace{14mu} p} \right\rbrack^{R_{B}}{mod}\mspace{14mu} p} \right\} \end{pmatrix}}} & (2) \end{matrix}$

Step 108: Communication entity B computes shared key K. The formula below is used to compute K:

$\begin{matrix} {K = {H_{5}\begin{pmatrix} {{A_{A}{B_{B}}{PW}{\frac{X}{H_{1}\left( {A_{A}{B_{B}}{PW}} \right)}}g^{R_{B}}{mod}\mspace{14mu} p}} \\ \left\{ {\left\lbrack \frac{X}{H_{1}\left( {A_{A}{B_{B}}{PW}} \right)} \right\rbrack^{R_{B}}{mod}\mspace{14mu} p} \right\} \end{pmatrix}}} \\ {= {H_{5}\left( {{A_{A}{B_{B}}{PW}{{g^{R_{A}}{mod}\mspace{14mu} p}}g^{R_{B}}{mod}\mspace{14mu} p}\left\{ {\left\lbrack g^{R_{B}} \right\rbrack^{R_{A}}{mod}\mspace{14mu} p} \right\}} \right)}} \end{matrix}$

Now communication entities A and B have authenticated the identity of each other and obtained the shared key K. That is, the exchange of shared key K is complete between communication entities A and B.

In the above steps, R_(A) and R_(B) are discrete logarithms that take g as the base modulo p. It is difficult to compute R_(A) and R_(B) based on g^(R) ^(A) mod p and g^(R) ^(B) mod p. In this case, R_(A) and R_(B) are private keys of communication entities A and B. X and Y are public keys of communication entities A and B. S₁ and S₂ are parameters for communication entities A and B to authenticate the identity of each other.

Though the security of the password-based key exchange authentication protocol can be proved in the random oracle model and ideal cipher model, transmission parameters X and Y are required in the authentication process and parameters X and Y are equal to the product of two large integers. Hence, the transmission of these parameters consumes large bandwidth. By analogy, in other occasions in which parameter transmission is required, when the parameter value is large, large transmission bandwidth may be consumed.

SUMMARY OF THE DISCLOSURE

The objective of the embodiments of the present disclosure is to provide a method and system for key exchange, a method and apparatus for reducing the parameter transmission bandwidth to reduce the bandwidth consumed during parameter transmission. The technical scheme of the present disclosure is implemented as follows. An embodiment of the present disclosure discloses a computer implemented method for key exchange, including: reducing, by a parameter sender, the values of transmission parameters by performing a modulo operation on a transmission parameter; and sending, by the parameter sender, the reduced value of the transmission parameter to a parameter receiver.

An embodiment of the present disclosure discloses a system for key exchange, including: a parameter sender, a parameter receiver and a bandwidth processing unit, wherein: the parameter sender is configured to send the transmission parameter to the bandwidth processing unit; the bandwidth processing unit is configured to perform a modulo operation on the received transmission parameter and send the transmission parameter to the parameter receiver.

An embodiment of the present disclosure discloses a computer implemented method for reducing the parameter transmission bandwidth, including: reducing, by a parameter sender, a value of a transmission parameter; sending, by the parameter sender, the reduced value of the transmission parameter to a parameter receiver.

An embodiment of the present disclosure also discloses an apparatus for reducing the parameter transmission bandwidth, wherein: the apparatus is configured to receive the parameter from a parameter sender, reduce a value of the parameter, and send the reduced value of the parameter to a parameter receiver.

Based on the preceding technical scheme of embodiments of the present disclosure, the values of the parameters are reduced before the parameters are sent to the parameter receiver. This reduces the bandwidth consumed during parameter transmission and makes the transmission more efficient.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows the process of key exchange as specified in the password-based key exchange authentication protocol in the prior art;

FIG. 2 shows the method for reducing the parameter transmission bandwidth in an embodiment of the present disclosure;

FIG. 3 shows the structure of an apparatus for reducing the parameter transmission bandwidth in an embodiment of the present disclosure;

FIG. 4 shows a method for key exchange in an embodiment of the present disclosure; and

FIG. 5 shows the structure of an apparatus for key exchange in an embodiment of the present disclosure.

DETAILED DESCRIPTION OF THE DISCLOSURE

The technical scheme in an embodiment of the present disclosure is described as follows: During the data transmission, the bandwidth consumed is reduced by reducing the values of the parameters so as to increase the transmission efficiency.

FIG. 2 shows a method for reducing the parameter transmission bandwidth in an embodiment of the present disclosure. As shown in FIG. 2, the method includes the following blocks:

Block 201: A parameter sender reduces the values of the parameters before sending the parameters to a parameter receiver.

Block 202: The parameter sender sends the reduced values of the parameters to the parameter receiver.

FIG. 3 shows the structure of an apparatus for reducing the parameter transmission bandwidth in an embodiment of the present disclosure. As shown in FIG. 3, an apparatus 301 for reducing the parameter transmission bandwidth includes a transceiver module 302 and a parameter processing module 303.

The transceiver module 302 is configured to receive the parameters from the parameter sender and send the parameters to the parameter processing module 303. Besides, the transceiver module 302 receives the parameters with reduced values from parameter processing module 303 and sends the parameters with reduced values to the parameter receiver.

The parameter processing module 303 is configured to receive the parameters from the transceiver module 302 and send back the parameters to the transceiver module 302 after reducing the values of the parameters.

The apparatus as shown in FIG. 3 may be integrated in the parameter sender or any third party system except the parameter sender and receiver. The third party system has other functions besides the functions shown in FIG. 3.

In the preceding method and apparatus for reducing the parameter transmission bandwidth, the values of the parameters can be reduced in the following ways: subtracting a constant from the parameters, multiplying the parameters by a number smaller than 1, or performing a modulo operation on the parameters.

The following describes the present disclosure in detail by using the transmission parameters specified in the password-based key exchange authentication protocol in combination with the accompanying drawings.

At least one of the transmission parameters as specified in the password-based key exchange authentication protocol is equal to the product of two large integers. Hence, the bandwidth consumed during parameter transmission is large. An embodiment of the present disclosure reduces the values of the transmission parameters by performing a modulo operation so as to develop a method for key exchange. This method can reduce the transmission width of the transmission parameters.

FIG. 4 shows the method for key exchange in an embodiment of the present disclosure. As shown in FIG. 4, the method includes the following blocks:

Block 401: Communication entity A selects a random number R_(A) to compute transmission parameter X=H₁(A_(A)|B_(B)|PW)·(g^(R) ^(A) mod p). Communication entity A obtains X′ after performing a modulo operation on transmission parameter X, and sends X′ to communication entity B as the transmission parameter. The formula below is used to compute X′:

X′=X mod p=[H ₁₍ A _(A) |B _(B) |PW)·g ^(R) ^(A] mod) p

In this block, because a modulo p operation is performed on X, the value of X′ is smaller than p. Besides, the value of Diffie-Hellman parameter p is smaller than X.

In this block, the modulo p operation on transmission parameter X may be performed by using an apparatus as shown in FIG. 3 for reducing the parameter transmission bandwidth. In this case, the transceiver module 302 receives parameter X from communication entity A which is the parameter sender, and sends parameter X to the parameter processing module 303. Besides, it receives parameter X′ from the parameter processing module 303 and sends parameter X′ to communication entity B which is the parameter receiver.

The parameter processing module 303 receives parameter X from the transceiver module 302, obtains X′ after performing a modulo operation on parameter X, and then sends X′ to the transceiver module 302.

As seen from the above, the parameter processing module 303 in this embodiment is a modulo operation module. The parameter value is reduced by performing a modulo operation on the parameter.

Block 402: Upon receipt of transmission parameter X′ from communication entity A, communication entity B checks whether X′ is 0. If yes, communication entity B terminates the process; otherwise communication entity B executes block 403.

Block 403: Communication entity B computes

$\frac{X^{\prime}}{H_{1}\left( {A_{A}{B_{B}}{PW}} \right)}{mod}\mspace{14mu} p$

to obtain g^(R) ^(A) mod p, and selects a random number R_(B) to compute transmission parameter Y and identity authentication parameter S₁′. Communication entity B then performs a modulo operation on transmission parameter Y to obtain transmission parameter Y′, and sends transmission parameter Y′ and identity authentication parameter S₁′ to communication entity A. The formulas below are used to compute Y, Y′ and S₁′.

$\begin{matrix} {{Y = {{H_{2}\left( {A_{A}{B_{B}}{PW}} \right)} \cdot \left( {g^{R_{B}}{mod}\mspace{14mu} p} \right)}}{Y^{\prime} = {{Y\mspace{11mu} {mod}\mspace{11mu} p} = {\left\lbrack {{H_{2}\left( {A_{A}{B_{B}}{PW}} \right)} \cdot g^{R_{B}}} \right\rbrack {mod}\mspace{14mu} p}}}} & \; \\ \begin{matrix} {S_{1}^{\prime} = {H_{3}\begin{pmatrix} {A_{A}{B_{B}}{PW}{{\frac{X^{\prime}}{H_{1}\left( {A_{A}{B_{B}}{PW}} \right)}{mod}\mspace{14mu} p}}g^{R_{B}}{mod}\mspace{14mu} p} \\ \left\{ {\left\lbrack \frac{X^{\prime}}{H_{1}\left( {A_{A}{B_{B}}{PW}} \right)} \right\rbrack^{R_{B}}{mod}\mspace{14mu} p} \right\} \end{pmatrix}}} \\ {= {H_{3}\begin{pmatrix} {A_{A}{B_{B}}{PW}{{g^{R_{A}}{mod}\mspace{20mu} p}}g^{R_{B}}{mod}\mspace{14mu} p} \\ \left\{ {\left\lbrack g^{R_{A}} \right\rbrack^{R_{B}}{mod}\mspace{14mu} p} \right\} \end{pmatrix}}} \end{matrix} & \; \end{matrix}$

In this block, because a modulo p operation is performed on Y, the value of Y′ is smaller than p. Besides, the value of Diffie-Hellman parameter p is smaller than Y.

In this block, the modulo p operation on transmission parameter Y may also be performed by using an apparatus as shown in FIG. 3 for reducing the parameter transmission bandwidth. In this case:

The transceiver module 302 receives parameter Y from communication entity B, which is the parameter sender, and sends parameter Y to the parameter processing module 303. Besides, the transceiver module 302 receives parameter Y′ from the parameter processing module 303 and sends parameter Y′ to communication entity A which is the parameter receiver.

The parameter processing module 303 receives parameter Y from the transceiver module 302, obtains Y′ after performing a modulo operation on parameter Y and then sends Y′ to the transceiver module 302.

As seen from above, the parameter processing module 303 in this embodiment is a modulo operation module. The parameter value is reduced by performing a modulo operation on the parameter.

Block 404: Upon receipt of transmission parameter Y′ from communication entity B, communication entity A checks whether Y′ is 0. If yes, communication entity A terminates the process; otherwise communication entity A executes block 405.

Block 405: Communication entity A checks whether the identity authentication parameter S₁′ sent from communication entity B is valid. If yes, communication entity A executes block 406; otherwise communication entity A terminates the process.

Communication entity A checks the identity authentication parameter S₁′ by the following method: compute

$\frac{Y^{\prime}}{H_{2}\left( {A_{A}{B_{B}}{PW}} \right)}{mod}\mspace{14mu} p$

to obtain g^(R) ^(B) mod p, and then check whether S₁′ is equal to the computation result of formula 3. If yes, S₁′ is valid; otherwise S₁′ is invalid. The following shows formula 3:

$\begin{matrix} {H_{3}\left( {{A_{A}{B_{B}}{PW}{{g^{R_{A}}{mod}\mspace{14mu} p}}\frac{Y^{\prime}}{H_{2}\left( {A_{A}{B_{B}}{PW}} \right)}{mod}\mspace{14mu} p\left. \left\{ {\left\lbrack \frac{Y^{\prime}}{H_{2}\left( {A_{A}{B_{B}}{PW}} \right)} \right\rbrack^{R_{A}} {mod}\mspace{20mu} p} \right\} \right)} = {H_{3}\left( {A_{A}{B_{B}}{PW}{{g^{R_{A}}{mod}\mspace{14mu} p}}g^{R_{B}}{mod}\mspace{14mu} p\left. \left\{ {\left\lbrack g^{R_{B}} \right\rbrack^{R_{A}}{mod}\mspace{14mu} p} \right\} \right)} \right.}} \right.} & (3) \end{matrix}$

Block 406: Communication entity A computes shared key K′ and identity authentication parameter S₂′ and sends S₂′ to communication entity B. The formulas below are used to compute K′ and S₂′.

$\begin{matrix} \begin{matrix} {K^{\prime} = {H_{5}\begin{pmatrix} {A_{A}{B_{B}}{PW}{{g^{R_{A}}{mod}\mspace{14mu} p}}\frac{Y^{\prime}}{H_{2}\left( {A_{A}{B_{B}}{PW}} \right)}{mod}\mspace{14mu} p} \\ \left\{ {\left\lbrack \frac{Y^{\prime}}{H_{2}\left( {A_{A}{B_{B}}{PW}} \right)} \right\rbrack^{R_{A}}{mod}\mspace{14mu} p} \right\} \end{pmatrix}}} \\ {= {H_{5}\begin{pmatrix} {A_{A}{B_{B}}{PW}{{g^{R_{A}}{mod}\mspace{14mu} p}}g^{R_{B}}{mod}\mspace{14mu} p} \\ \left\{ {\left\lbrack g^{R_{B}} \right\rbrack^{R_{A}}{mod}\mspace{14mu} p} \right\} \end{pmatrix}}} \end{matrix} \\ \begin{matrix} {S_{2}^{\prime} = {H_{4}\begin{pmatrix} {A_{A}{B_{B}}{PW}{{g^{R_{A}}{mod}\mspace{14mu} p}}\frac{Y^{\prime}}{H_{2}\left( {A_{A}{B_{B}}{PW}} \right)}{mod}\mspace{14mu} p} \\ \left\{ {\left\lbrack \frac{Y^{\prime}}{H_{2}\left( {A_{A}{B_{B}}{PW}} \right)} \right\rbrack^{R_{A}}{mod}\mspace{14mu} p} \right\} \end{pmatrix}}} \\ {= {H_{4}\begin{pmatrix} {A_{A}{B_{B}}{PW}{{g^{R_{A}}{mod}\mspace{14mu} p}}g^{R_{B}}{mod}\mspace{14mu} p} \\ \left\{ {\left\lbrack g^{R_{B}} \right\rbrack^{R_{A}}{mod}\mspace{14mu} p} \right\} \end{pmatrix}}} \end{matrix} \end{matrix}$

Block 407: Communication entity B checks whether the identity authentication parameter S₂′ sent from communication entity A is valid. If yes, communication entity B executes block 408; otherwise communication entity B terminates the process.

Communication entity B checks identity authentication parameter S₂′ by the following method: obtain g^(R) ^(B) mod p according to block 405, and then check whether S₂′ is equal to the computation result of formula 4. If yes, S₂′ is valid; otherwise S₂′ is invalid. The following shows formula 4:

$\begin{matrix} {\left. {{H_{4}\left( {A_{A}{B_{B}}{PW}} \right.}\frac{X^{\prime}}{H_{1}\left( {A_{A}{B_{B}}{PW}} \right)}{mod}\mspace{14mu} p{{g^{R_{B}}{mod}\mspace{14mu} p}}\left\{ {\left\lbrack \frac{X^{\prime}}{H_{1}\left( {A_{A}{B_{B}}{PW}} \right)} \right\rbrack^{R_{B}} {mod}\mspace{20mu} p} \right\}} \right) = {H_{4}\left( {A_{A}{B_{B}}{PW}{{g^{R_{A}}{mod}\mspace{14mu} p}}g^{R_{B}}{mod}\mspace{14mu} p\left. \left\{ {\left\lbrack g^{R_{B}} \right\rbrack^{R_{A}}{mod}\mspace{14mu} p} \right\} \right)} \right.}} & (4) \end{matrix}$

Block 408: Communication entity B computes shared key K′. The formula below is used to compute K′:

$\begin{matrix} {K^{\prime} = {H_{5}\begin{pmatrix} {{A_{A}{B_{B}}{PW}{{\frac{X^{\prime}}{H_{1}\left( {A_{A}{B_{B}}{PW}} \right)}{mod}\mspace{14mu} p}}g^{R_{B}}{mod}\mspace{14mu} p}} \\ \left\{ {\left\lbrack \frac{X^{\prime}}{H_{1}\left( {A_{A}{B_{B}}{PW}} \right)} \right\rbrack^{R_{B}}{mod}\mspace{14mu} p} \right\} \end{pmatrix}}} \\ {= {H_{5}\begin{pmatrix} {A_{A}{B_{B}}{PW}{{g^{R_{A}}{mod}\mspace{14mu} p}}g^{R_{B}}{mod}\mspace{14mu} p} \\ \left\{ {\left\lbrack g^{R_{B}} \right\rbrack^{R_{A}}{mod}\mspace{14mu} p} \right\} \end{pmatrix}}} \end{matrix}$

Now the exchange of shared key K′ between communication entities A and B is complete.

In a method for key exchange in the preceding embodiment, a modulo operation is performed to reduce X and Y to X′ and Y′ respectively. This reduces the transmission bandwidth of transmission parameters compared with the password-based key exchange authentication protocol in the prior art.

Similarly, in other occasions in which parameter transmission is required, the method for reducing the bandwidth consumed during parameter transmission by reducing the parameter values is also applicable. The method for reducing the parameter values includes: subtracting a constant from the parameters, or multiplying the parameters by a number smaller than 1, or performing a modulo operation on the parameters as provided in an embodiment of the present disclosure.

When a modulo operation is performed to reduce the values of the transmission parameters, the formulas for computing the transmission parameters include Hash operation formula, or modulo index operation formula, or Hash operation formula and modulo index operation formula. Here the modulo index operation formula refers to the operation formula in which a modulo operation is performed on the index. g^(R) ^(A) mod p in FIG. 4 is such an example.

Based on the above embodiment, the following describes a system for key exchange as shown in FIG. 5.

FIG. 5 shows the structure of a system for key exchange in an embodiment of the present disclosure. As shown in FIG. 5, the system includes a parameter sender 501, a parameter receiver 502 and a bandwidth processing unit 503. The parameter sender 502 is configured to send the transmission parameters to the bandwidth processing unit 503. The bandwidth processing unit 503 is configured to perform a modulo operation on the received transmission parameters and then send the reduced values of the transmission parameters to the parameter receiver 502.

In FIG. 5, the parameter sender 501 includes a parameter generating unit 504. The parameter generating unit 504 is configured to generate transmission parameters for Hash operation formula and/or modulo index operation formula and send the generated transmission parameters to the bandwidth processing unit 503.

In FIG. 5, the parameter sender 501 is configured to send the transmission parameters of the key exchange method as specified in the password-based key exchange authentication protocol, to the bandwidth processing unit 503.

The bandwidth processing unit 503 may be integrated in the parameter sender 501 or any third party equipment except the parameter sender 501 and parameter receiver 502.

Although the disclosure has been described through some exemplary embodiments, the disclosure is not limited to such embodiments. It is apparent that those skilled in the art can make various modifications and variations to the disclosure without departing from the spirit and scope of the disclosure. The disclosure is intended to cover the modifications and variations provided that they fall in the scope of protection defined by the following claims or their equivalents. 

1. A computer implemented method for key exchange, comprising: reducing, by a parameter sender, a value of a transmission parameter by performing an operation on the transmission parameter; sending, by the parameter sender, the reduced value of the transmission parameter to a parameter receiver.
 2. The computer implemented method of claim 1, wherein the operation on the transmission parameter to reduce the value of the transmission parameter comprises at least one of a Hash operation and a modulo index operation.
 3. The method of claim 2, wherein the computer implemented method is based on an password-based key exchange authentication protocol.
 4. The computer implemented method of claim 2, wherein the transmission parameter is calculated using: X=H₁(A_(A)|B_(B)|PW)·(g^(R) ^(A) mod p); wherein the modulo operation is implemented by X′=X mod p=[H ₁(A _(A) |B _(B) |PW)·g ^(R) ^(A) ] mod p; wherein X is the transmission parameter; X′ is the reduced value of the transmission parameter; A_(A) and B_(A) are parameters belonging to the parameter sender and the parameter receiver respectively and are known to the parameter sender and the parameter receiver; PW is a secret password shared by the parameter sender and the parameter receiver; R_(A) is a private key of the parameter sender; p is a prime number; g is a positive integer; | is a concatenation symbol; mod is the modulo operator; H₁ (•) indicates the Hash function.
 5. The computer implemented method of claim 4, further comprising: receiving, by the parameter receiver, the reduced value of the transmission parameter X′ from the parameter sender; computing $\frac{X^{\prime}}{H_{1}\left( {A_{A}{B_{B}}{PW}} \right)}{mod}\mspace{14mu} p$ to obtain g^(R) ^(A) mod p.
 6. A system for key exchange, comprising: a parameter sender, a parameter receiver, and a bandwidth processing unit, wherein: the parameter sender is configured to send a transmission parameter to the bandwidth processing unit; the bandwidth processing unit is configured to perform a modulo operation on the received transmission parameter and send the transmission parameter to the parameter receiver.
 7. The system of claim 6, wherein the parameter sender comprises a parameter generating unit configured to generate the transmission parameter for a Hash operation or a modulo index operation and send the generated transmission parameter to the bandwidth processing unit.
 8. The system of claim 6, wherein the parameter sender is configured to send the transmission parameter for a key exchange method based on a password-based key exchange authentication protocol to the bandwidth processing unit.
 9. The system of claim 7, wherein the parameter sender is configured to send the transmission parameter for a key exchange method based on a password-based key exchange authentication protocol to the bandwidth processing unit.
 10. The system of claim 6, wherein the bandwidth processing unit is integrated in the parameter sender, or an third party equipment except the parameter sender and the parameter receiver.
 11. The system of claim 7, wherein the bandwidth processing unit is integrated in the parameter sender, or an third party equipment except the parameter sender and the parameter receiver.
 12. A computer implemented method for reducing parameter transmission bandwidth, comprising: reducing, by a parameter sender, a value of a transmission parameter; sending, by the parameter sender, the reduced value of the transmission parameter to a parameter receiver.
 13. The computer implemented method of claim 12, wherein the step of reducing the value of the transmission parameter comprises: performing a modulo operation on the transmission parameter; or subtracting a constant from the transmission parameter; or multiplying the transmission parameter by a number smaller than
 1. 14. An apparatus for reducing parameter transmission bandwidth, wherein the apparatus is configured to receive a parameter from a parameter sender, to reduce a value of the parameter, and to send the reduced value of the parameter to a parameter receiver.
 15. The apparatus of claim 14, comprising: a transceiver module and a parameter processing module, wherein: the transceiver module is configured to receive the parameter from the parameter sender, send the parameter to the parameter processing module, to receive a reduced value of the parameter from the parameter processing module, and send the reduced value of the parameter to the parameter receiver; and the parameter processing module is configured to receive the parameter from the transceiver module and send back the reduced value of the parameter to the transceiver module after reducing the value of the parameter. 